So I am a (former) quantum physicist (though the wrong kind, an experimental–elementary-particles type rather than a theoretical–condensed-matter type) and am now an electrical engineer, and I can say with confidence, bipolar junction transistors are hard to really understand. I don't think I've ever seen a single explanation that goes through it all in an understandable way.
In particular, factors that have to be considered are the built-in potentials of the space charge layers (aka depletion zones), and the different doping profile of the collector versus emitter. The doping is very important, and is often neglected. But without it, transistors would basically operate equally well in forward mode as in reverse, and they definitely don't do that. These two things have a big effect on the Fermi/band-bend diagram and probably help clear up your last point.
I barely feel like I actually understand these things on a good day, and I pay the bills with them....
QM isn't too bad. The basic ideas are alien but straightforward enough. It's hard to extend small examples to larger systems, though, and that plus the innate weirdness really messes with your intuition.
BJTs are complicated devices built from multiple simple-enough effects all happening at the same time. Each piece isn't too bad, but you've got to track them all and combine them into an overall function.
QFT (quantum field theory, such as QED) is basically all the bad parts of both of those: the basics aren't too bad, but it's difficult to scale, there's a lot going on, and it's hard to develop intuition. QCD then makes the basic calculations intractable, which is why it's still poorly understood to this day (in my opinion).
So, yeah, in conclusion, complexity is complex? Not so profound, I guess.
Yet Shockley was able to make quantitative predictions about, say, solar cell performance that are still considered useful today. And I don't think kT shows up en the Ebers–Moll model just by chance.
In particular, factors that have to be considered are the built-in potentials of the space charge layers (aka depletion zones), and the different doping profile of the collector versus emitter. The doping is very important, and is often neglected. But without it, transistors would basically operate equally well in forward mode as in reverse, and they definitely don't do that. These two things have a big effect on the Fermi/band-bend diagram and probably help clear up your last point.
I barely feel like I actually understand these things on a good day, and I pay the bills with them....